Feynman Path Integral Formulation

7.3 Lattice Diffeomorphism Invariance 239
l 2 i = l 2 0i + q i + δl 2 i , (7.49)
where q i describes an arbitrary but small deviation from a regular lattice, and δl 2 i is
a gauge fluctuation, whose form needs to be determined. We shall keep terms O(q 2 )
and O(q δl 2 ), but neglect terms O(δl 4 ).
The squared volumes V 2 n (σ) of n-dimensional simplices σ are given by homogeneous
polynomials of order (l 2 ) n . In particular for the area of a triangle A Δ with
arbitrary edges l 1 ,l 2 ,l 3 one has
δA 2 Δ = 1 8 (−l2 1 + l 2 2 + l 2 3)δl 2 1 + 1 8 (l2 1 − l 2 2 + l 2 3)δl 2 2 + 1 8 (l2 1 + l 2 2 − l 2 3)δl 2 3 , (7.50)
and similarly for the other quantities which are needed in order to construct the
action. For our notation in two dimensions we refer to Fig. (7.5). The subsequent
Figs. 7.6 and 7.7 illustrate the difference between a gauge deformation of the surface,
which leaves the area and curvature at the point labeled by 0 invariant, and a
physical deformation which corresponds to a re-assignment of edge lengths meeting
at the vertex 0 such that it alters the area and curvature at 0. In the following we will
characterize unambiguously what we mean by the two different operations.
Consider therefore an expansion about a deformed equilateral lattice, for which
l 0i = 1 to start with. A motivation for this choice is provided by the fact that in
the numerical studies of two-dimensional gravity the averages of the squared edge
lengths in the three principal directions turn out to be equal, 〈l1 2〉 = 〈l2 2 〉 = 〈l2 3 〉.The
baricentric area associated with vertex 0 is then given by
A = A 0 (q)+ 1 [
2 · 3 5/2 δl01 2 (3 + q 06 − 4q 01 + q 02 + q 16 + q 12 )
+ δl02 2 (3 + q 01 − 4q 02 + q 03 + q 12 + q 23 )
+ δl03 2 (3 + q 02 − 4q 03 + q 04 + q 23 + q 34 )
+ δl04 2 (3 + q 03 − 4q 04 + q 05 + q 34 + q 45 )
+ δl05 2 (3 + q 04 − 4q 05 + q 06 + q 45 + q 56 )
]
+ δl06 2 (3 + q 05 − 4q 06 + q 01 + q 56 + q 16 )
+ O(δl 4 ) .
(7.51)
The normalization here is such that A 0 =
in more compact notation, at the vertex 0
√
3
2 for q i = 0. Equivalently one can write,
A = A 0 (q)+ 1 3 v A(q) · δl 2 + O(δl 4 ) , (7.52)
with δl 2 =(δl01 2 ,...,δl2 06
). After adding the contributions from the neighboring
vertices one obtains
∑ A = ∑ A 0 (q)+v A (q) · δl 2 + O(δl 4 ) . (7.53)
P 0 ...P 6 P 0 ...P 6